Gravity and Strings

7.2 Sources for Schwarzschild’s solution 201
There are several reasons why we can expect a negative result: first of all, if the source
for the Schwarzschild field were a massive point-particle, it would give rise to a timelike
singularity along its worldline, but we know that the Schwarzschild singularity is spacelike.
Second, the source for the gravitational field is not just mass, but any kind of energy,
including the gravitational field itself. Thus, even if we have a mass distribution confined
to a finite region of space (in an idealized case, a point), the gravitational field that it generates
will fill the whole space and the source (mass and field) will not be confined to that
region. In a sense this is already taken care of by Einstein’s equations: in our construction
of the self-consistent spin-2 theory we saw that the Einstein tensor contains the “gravitational
energy–momentum (pseudo)tensor” and only the matter sources are on the r.h.s. of
Einstein’s equations.
Anyway, we are going to check explicitly that the massive point-particle cannot be the
source for the Schwarzschild metric. This calculation will prepare us for future calculations
of the same kind, which, in contrast, will be successful and will help us to understand the
reason why.
We consider the action for a massive particle coupled to gravity (we ignore boundary
terms):
S[gµν, X µ (ξ)] =
c3 16πG (4)
N
d 4 x
|g| R − Mc
The equations of motion of gµν(x) and X µ (ξ) are, respectively,
where
Gµν(x) +
8π MG(4)
N c−2
√
|g|
dξ gµρ(X)gνσ(X) ˙X ρ ˙X σ
|gλτ (X) ˙X λ ˙X τ |
dξ
|gµν(X) ˙X µ ˙X ν |. (7.32)
δ (4) [X (ξ) − x] = 0, (7.33)
γ 1 2 M∇ 2 (γ )X λ + Mγ − 1 2 Ɣρσ λ ˙X ρ ˙X σ = 0, (7.34)
γ = gµν(X) ˙X µ ˙X ν . (7.35)
In the physical system that we are considering, the Schwarzschild gravitational field is
produced by a point-particle that is at rest in the frame that we are going to use (Schwarzschild
coordinates). Then, we expect the solution for X µ (ξ) to be
X µ (ξ) = δ µ 0ξ. (7.36)
However, the X µ equations of motion are not satisfied because the component Ɣ00 r does
not vanish at the origin. Actually, it diverges, and we face here the problem of the infinite
force that the gravitational field exerts over the source itself, which is similar to the infiniteself-energy
problem of the classical electron mentioned at the beginning of this section.
We will see that, in certain situations (in the presence of unbroken supersymmetry), this
problem does not occur because the divergent gravitational field is canceled out by another
divergent field (electromagnetic, scalar ...) and the equation of motion of the particle (or
brane) canbe solved exactly.